High School: Geometry
2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
When it comes to circles, students often think that they don't have to worry about angles anymore. After all, a circle is a curve. No corners means no angles, right? Right?
You may have to break the news to them gently because actually, circles have more angles than an angler fish. (Note: the precise number of angles in an angler fish has yet to be determined.)
Before students can identify and describe the various angles in a circle, they should be familiar with what these angles are. Students should also be familiar with the concept of arc measurement and how it relates to the measure of the different kinds of angles. Throw some chords in there, too, and we aren't talking about your guitar skills (though we're sure you can rock and roll with the best of 'em).
Students should know that a central angle is formed by two radii (where the vertex of the angle is the center of the circle), an inscribed angle is an angle formed by two chords (where the vertex of the angle is some point on the circle), and a circumscribed angle has a vertex outside the circle and sides that intersect with the circle.
To solidify the relationships between arc measure and angle measure, we suggest bringing a pizza to class. Explain that usually, we cut pizzas so that each slice of pizza has a particular central angle. The amount of crust (arc measure) depends on the central angle of the pizza slice.
If you have the whole pizza to yourself, you could even cut a huge slice from one side to the other, resulting in an inscribed angle. The measure of that angle will also affect how much crust you get on your slice.
Somewhere in between the discussions of angles and pizza toppings, throw in the formulas for finding the measures of inscribed and circumscribed angles when given the central angle or arc measure. Most important of these is that an inscribed angle's measure is equal to half the measure of its intercepted arc.
After students have mastered these angles and arc measures, talk about tangents and secants of circles, lines that intersect a circle at one and two points, respectively. It's also important for students to know that a point on a circle can only have one tangent and that tangents are always perpendicular to the radius of a circle.
Once you're done explaining all these relationships, feel free to gulp the lesson down with a big slice of pepperoni. Bust out your guitar chops while you're at it, and have a party.